 reserve R for finite Approximation_Space;
 reserve X,Y,Z for Subset of R;
 reserve kap for RIF of R;

theorem :: Proposition 7 aL)
  (delta_L R).(X,Y) = 0 iff X = Y
  proof
B1: (CMap kappa R).(X,Y) >= 0 & (CMap kappa R).(Y,X) >= 0 by XXREAL_1:1;
    hereby assume (delta_L R).(X,Y) = 0; then
      ((CMap kappa R).(X,Y) + (CMap kappa R).(Y,X)) / 2 = 0 by DeltaL; then
      (CMap kappa R).(X,Y) = 0 & (CMap kappa R).(Y,X) = 0 by B1;
      hence X = Y by Prop6a;
    end;
    assume
A1: X = Y;
    (delta_L R).(X,Y) = ((CMap kappa R).(X,Y) + (CMap kappa R).(Y,X)) / 2
      by DeltaL
       .= (0 + 0) / 2 by Prop6a,A1;
    hence thesis;
  end;
